91Ó°ÊÓ

Damped harmonic motion is a fascinating phenomenon where oscillations decrease over time, typically due to friction or other resistance forces. An example of damped harmonic motion can be observed in a swinging pendulum that gradually comes to a stop.

When analyzing damped harmonic motions mathematically, we often use a differential equation. This equation typically has the form:

• \( \ddot{x}(t) + 2\mu \dot{x}(t) + \omega_{0}^{2} x(t) = 0 \)

Here, \( \mu \) is the damping coefficient, which represents how strong the damping force is. The term \( \omega_0 \) is the natural frequency of the system, describing how it would oscillate without damping.

Depending on the relationship between \( \mu \) and \( \omega_0 \), the system can exhibit different types of damping:

• Underdamped: \( \mu^2 < \omega_0^2 \), where oscillations gradually decrease over time.
• Critically Damped: \( \mu^2 = \omega_0^2 \), where the system returns to equilibrium as fast as possible without oscillating.
• Overdamped: \( \mu^2 > \omega_0^2 \), where the system returns to equilibrium without oscillating and slower than critical damping.

In dealing with non-homogeneous differential equations, finding a particular solution is crucial. It represents the specific response of the system to an external force. For a damped harmonic oscillator with an external force \( b \sin(\Omega t) \), we seek a particular solution.A method to find this involves proposing a solution of the form:

• \( x_p(t) = A \sin(\Omega t) + B \cos(\Omega t) \)

The coefficients \( A \) and \( B \) are determined by substituting this proposed solution into the differential equation and comparing it to the external force term. Often, one component (either \( A \) or \( B \)) will drop out because the forced oscillation matches in phase with one of the trigonometric functions.

In this solved example, after comparison, we find that typically \( B = 0 \), aligning with the nature of the forcing term being purely sine. The value of \( A \) is then calculated with:

• \( A = \frac{b}{\omega_0^2 - \Omega^2 + 2i\mu\Omega} \)

This expression captures how the system uniquely responds to the external force frequency \( \Omega \).

To solve a differential equation like our damped harmonic oscillator, we first find the homogeneous solution. This represents the intrinsic behavior of the system without any external influence.The homogeneous solution arises from solving the corresponding homogeneous equation:

• \( \ddot{x}(t) + 2\mu \dot{x}(t) + \omega_{0}^{2} x(t) = 0 \)

Solving this involves finding the roots of the characteristic equation:

• \( r^2 + 2\mu r + \omega_{0}^{2} = 0 \)

Depending on the damping, we find real or complex roots. In an underdamped scenario, these roots are complex:

• \( r = -\mu \pm i\sqrt{\omega_{0}^{2} - \mu^{2}} \)

This leads to a homogeneous solution of:

• \( x_h(t) = e^{-\mu t} (C_1 \cos(\omega_d t) + C_2 \sin(\omega_d t)) \)

where \( \omega_d = \sqrt{\omega_{0}^{2} - \mu^{2}} \) is the damped natural frequency. The constants \( C_1 \) and \( C_2 \) are determined by specific initial conditions, uniquely tailoring the solution to fit the initial state of the system.